I am an engineering student looking to get into using Shapeways. All my previous experience is with Autodesk Inventor, but after seeing some of the creations on this site and tutorials for software like Blender and Meshlab, I'm having second doubts about Inventor's practicality and power for this sort of modeling. What are people's experiences with using engineering-oriented vs. art-oriented software for Shapeways?

I think it depends on the kind of models you want to create.
Parametric CAD modelers, such as Inventor are very good for modeling "mechanical" parts, but usually fall short on modeling free-form shapes, such as found in organic elements.
I am an engineer and my professional experience is with CATIA, which is also an engineering tool. I prefer parametric modeling, but again, the kind of parts I make are more mechanical than "organic".

Many designs for 3D printing have very detailed geometry that would be hard to work with in a CAD program. Detail is essentially free to print, and printing is charged by volume, which encourages more complicated shapes. Precision is also much more limited, so typing in exact values is less useful than having rapid visual feedback, and mesh modelers are better with this. Since printed models are often unique creations that don't need to be heavily modified, the value of parametric modeling is reduced.

You will probably need the precision of CAD for some projects, but if you want to quickly familiarize yourself with the kind of modeling that takes the most advantage of it's 3D printing, start with TopMod. It's like a Cambrian explosion of geometric forms. Much of what you see in the gallery is topological modeling.

Well, I use AutoCAD for my stuff, which is mechanically-oriented (transforming robots and stuff), and it works a treat, though I'm experimenting with Blender and TrueSpace to make some more organic detailing.

As Glehn said, it depends on what you want to model, but whether you're aiming to make technical pieces in AutoCAD or Inventor, or more arty organic works in Blender or ZBrush, they're all just fine for printing.